3. Dynamic Response of Systems Equipped with Viscous and Viscoelastic Dampers

In this chapter, the principal elements of the dynamic response of linear undamped and damped single and multi degrees-of-freedom systems are described in time domain both in terms of the relative motions of the mass and in state space form. In a first stage, the analysis of the dynamic response of linear simple-degree-of-freedom systems in the time domain is described by defining the main dynamic characteristics of the undamped system and, then, evaluating the damped natural frequency of the system and response factor. The modal strain energy method is described to evaluate the damping ratio of the SDOF system equipped with viscous and viscoelastic devices. As for linear multi-degrees-of-freedom systems the decoupling procedure for modal analysis and the proportional damping model are described. Even for these systems the effect of damping, assumed to be proportional, on the natural frequencies is evaluated and the modal strain energy method with regard to viscous and viscoelastic devices is illustrated. Then, the concept of the state of a system, the definition of the state space and its properties are discussed. Finally, with reference to both single-degree-of-freedom that multi-degrees-of-freedom systems the representation of their dynamic response in state space is illustrated.